<p>In this paper, we propose a stochastic single projection method for solving a stochastic variational inequality problem. Double inertial extrapolation steps are used to accelerate the convergence speed, only one projection needs to be computed, and a self-adaptive step size is incorporated for dealing with the unknown Lipschitz constant of the mapping in our method. Under the moderate conditions we prove the convergence and discuss the convergence rate of the proposed algorithm. Furthermore, we also investigate the linear convergence and complexity of our algorithm under some suitable assumptions. Some numerical examples are given to illustrate the effectiveness of our algorithm and compare the numerical results with some related algorithms. The numerical results show that our algorithm is more competitive.</p>